Estimating fractal dimension with the divider method in geomorphology

Geornorphology, 5 (1992) 131-141 Elsevier Science Publishers B.V., Amsterdam

131

Estimating fractal dimension with the divider method in

geomorphology Robert Andrle Department of Geography U-148, Universityof Connecticut, Storrs, CT 06269, USA (Received April 6, 1991; accepted after revision November 4, 1991 )

ABSTRACT Andrle, R., 1992. Estimating fractal dimension with the divider method in geomorphology. In: R.S. Snow and L. Mayer (Editors), Fractals in Geomorphology. Geomorphology, 5:131-141. In order to investigate sources of error in using the divider method to estimate the fractal dimension of geomorphic phenomena, the divider method is applied to two river channel traces, a topographic contour line and a coastline. The lines were chosen to represent a variety of types of geomorphic phenomena. Three sources of error are examined. These include the problem of the last partial step in a divider walk, of varying the starting point of a divider walk, and of nonlinearity in the relationship between measured length and steplength, and/or number of steps and steplength. The amount of error in estimates of fractal dimension that results from each of these sources is difficult to determine. Procedures that can reduce the error are identified; however, error from these sources cannot be eliminated entirely. Therefore, it is suggested that researchers use greater care when employing the divider method to produce estimates of fractal dimension for use in geomorphic analyses.

Introduction

In applying the divider method to talus slope surfaces, Andrle and Abrahams (1989) found that apparently linear log-log plots of steplength G against number of steps N exhibited systematic curvature. The evidence revealed in their study brings into question the reliability of the divider method as it is commonly used to estimate fractal dimension D. However, the data used in their study was limited and further evidence is needed. In this study, the divider method is applied to four geomorphic lines differing in character. The primary goal of this study is to test the ability of the divider method to produce reliable estimates of D for geomorphic phenomena. Three sources of erCorrespondence to: R. Andrle, Department of Geography U- 148, University of Connecticut, Storrs, CT 06269, USA.

ror in estimating D are examined: (1) the problem of the last partial step; (2) the effect of varying the starting point of a divider walk; and (3) the effect of nonlinearity in the relationship between measured length and steplength, and/or number of steps and steplength, both of which are used to derive estimates of D. The divider method and fractal dimension

The divider method is based on a technique traditionally employed by cartographers to measure the length of irregular lines on maps. To implement this technique the investigator walks a pair of dividers set at a fixed steplength G along a line and counts the number of steps N taken. The measured length L (G) of the line is equal to the product of N and G. In order to

0169-555X/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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calculate the fractal dimension of a line, values of N must be obtained for at least two values of G. The fractal dimension D of a line can be derived from the equation (Mandelbrot, 1967):

L ( G ) = M G '-D

(1)

where L(G) is an estimate of the length of a line measured with a unit of length G, and M is a positive constant. For a line to be a fractal, D must exceed 1, indicating that as steplength G decreases, measured length L (G) increases as a power function of G. The larger the value of D, the more complex the line. As D approaches 2, the line becomes so complex that it nearly fills the plane. A line that has a constant value of D over a range of scale is said to be -statistically self-similar (Mandelbrot, 1967), which implies that the line's relative complexity is independent of the scale of measurement over that range of scale. One method commonly employed to estimate D involves plotting values of L(G) against G on log-log paper. This type of plot is called a Richardson plot (after Richardson, 1961 ). If the Richardson plot appears linear, a best-fit line is then drawn through the scatter of points either visually or by the use of leastsquares linear regression. D is equal to 1 minus the slope of the line. D can also be estimated using values of N and G. If only two steplengths are used, D can be calculated from the equation (Goodchild, 1980; Mark, 1984; Goodchild and Mark, 1987):

D=Iog(N2/N, ) log( G1/G2)

(2)

where N1 and N2 are the number of steps of lengths GI and G2 (G1 > G2) needed to span a section of a line. If more than two steplengths are employed, D is usually estimated from a plot of log N against log G. As with the Richardson plot, the usual procedure employed with the log N-log G plot is to examine the plot visually for linearity. If the plot appears linear, a best-fit line is then drawn through the scatter

either by eye or by least-squares linear regression analysis. D is equal to the absolute value of the slope of the relation.

Fractal elements More recently, variations in fractal dimension over scale have been observed for some geomorphic phenomena. This has led to the development of the fractal element model (Orford and Whalley, 1983). In this model, ranges of scale called fractal elements, within which D is constant, are separated by transition zones in which D varies. It is generally accepted that each fractal element is produced by a single process and corresponds to a range of scale over which the process dominates the formation of the landscape (Orford and WhalIcy, 1983; Mark and Aronson, 1984). Within each fractal element the line is believed to exhibit statistical self-similarity and, hence, can be represented accurately by a single value of D.

Methodology A computer program implementing an interpolating divider walk was written by the author for use in this analysis. Similar divider programs have been used by other researchers in the past (e.g., Shelberg et al., 1983). Two input files are first created for use by the program, one containing the set of G values to be used, and another containing a list of digitized coordinates of the line to be studied. The program outputs values of G, N and L ( G ) , which are then used to derive estimates of D for each line. Four geomorphic lines which differ in character were digitized for use in the subsequent analyses (Fig. 1 ). Included are channel traces of two meandering rivers, a topographic contour line and a coastline. Antietam Creek is a meandering river located west of Hagerstown, Maryland, and the section is digitized from a 1 : 50,000 scale Army Map Service topographic

ESTIMATING FRACTAL DIMENSION WITH THE DIVIDER METHOD IN GEOMORPHOLOGY

133

one-half the length of the smallest steplength used in the analysis.

Analysis The last partial step

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INDANAO COASTLINE

Fig. 1. Outlines of the four digitized lines.

map (Hagerstown, Maryland Quadrangle). The Little Missouri River, North Dakota, is a tributary of the Missouri River and was digitized from a 1:250,000 scale U.S. Geological Survey map (Dickinson, North Dakota map sheet). The Grand Canyon contour line is a section of the 4000 ft contour line on the north side of the gorge taken from a 1 : 48,000 scale U.S. Geological Survey map (Bright Angel Quadrangle), and the coastline of Mindanao is an irregular volcanic island coastline digitized from a 1:250,000 scale Army Map Service topographic map (Caraga, Philippine Islands map sheet). The smallest steplengths used in the following analyses are larger than the level of cartographic generalization on the source maps. The lines were digitized using an incremental track mode which produced a consistent point spacing. In order to minimize the effect of digitization on the analysis, all four lines were digitized at a point spacing no greater than

One source of error in estimating D with the divider method centers on the problem of accounting for the last partial step in a divider walk which invariably occurs at the end of a line being measured. Three possible solutions are suggested. First, the last partial step can be ignored. Second, the last partial step can be rounded off (i.e., another whole step is added if the last partial step is greater than G/2). Third, the distance to the endpoint of the digitized line can be added to the total number of steps (expressed as a fraction of G). In all three cases the problem of determining the exact amount of error is perplexing because the amount of error is dependent upon the configuration of the line extending beyond the endpoint of the digitized section. Therefore, rigorous comparison of the three solutions is difficult, and apparently no consensus regarding this issue has yet been reached in the literature. It is evident that for the first solution (ignoring the last partial step) the maximum possible error in estimating L(G) approaches G. Therefore, this does not appear to be the best solution. However, it can be shown that in an extreme case such as that shown in Fig. 2, the maximum possible error for the other two solutions also approaches G. If the divider walk continued, the next divider step should occur at point C, overstepping the intervening bend which is smaller than G. Rounding the last partial step (length AB in Fig. 2), where AB> G/2, results in the addition of a whole step and, hence, in the overestimation ofL (G) by one steplength G. Adding the last partial step as a fraction of G in this example will also result in an overestimation of L (G). However, for this solution the error will be equal not to

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\

Fig. 2. Explanation of the effect of the last partial step on error in estimating measured length L(G). The solid curved line is a portion of the digitized section of line, and the dotted line is a hypothetical extension of the same line beyond the last digitized point. A is the endpoint of the last whole divider step, B is the endpoint of the digitized line, and C is the hypothetical position of the next whole divider step point.

G, but to the length of the last partial step AB. Thus it will tend to produce a somewhat smaller error in L(G) (and, hence, D). Extreme cases such as that shown in Fig. 2, while possible, do not seem likely to be encountered with great frequency. A more useful examination of the three solutions described above can be conducted by studying more likely situations (i.e., comparing the mean error from a large number of divider walks rather than using the m a x i m u m possible error). Toward this end, approximately 600 divider walks using different starting points were conducted on the four lines shown in Fig. 1 and the length of the last partial step of each walk computed. The lengths of the last partial steps from the 600 walks approximated a uniform distribution with a mean of G/2. Even with this knowledge, a precise comparison of the error resulting from the three solutions is impossible without knowing the configuration of the line beyond the endpoint of the digitized section. However, assuming that extreme line configurations are unlikely, the following discussion might provide some insight into the problem. Given that the lengths of the last partial steps are uniformly distributed, as might be expected the first solution (ignoring the last par-

tial step) would appear to be the worst. In all cases (where the last partial step ranges uniformly from between 0 and G), no steplength is added, and mean error would be approximately G/2. The second solution (rounding), provides a closer approximation and would reduce mean error approximately in half by reducing the disparity between the actual length of the last partial step and the distance selected (in this case, either G = 0, or G = 1 ). The third solution (adding the last partial step as a fraction of G), by similar reasoning, ought to provide an even closer approximation and, hence, reduce error further. Clearly this mostly qualitative discussion is not conclusive and the problem needs more rigorous examination. The primary difficulty in conducting a more detailed study of the problem hinges on incorporating the configuration of the line beyond the endpoint of the digitized section into the analysis. Lacking more conclusive information, the divider program used in this study adds the last partial step as a fraction of G. Regardless of which solution is chosen, the error resulting from the last partial step cannot be eliminated entirely and its effect on estimating D can be noted. As G increases in proportion to L ( G ) , error in estimating D increases because the error resulting from the last partial step increases along with G. One means of eliminating this error entirely was employed by Richardson ( 1961 ) in which the only steplengths used are those which reach the endpoint of the line in a whole number of steps. However, this solution is difficult to implement and it severely reduces the number of steplengths available for use. A more practical solution is to limit the m a x i m u m size of steplength used in the analysis. Unfortunately this restricts the useful range of scale of measurement.

The startingpoint problem In his examination of the length of national boundaries, Richardson ( 1961 ) noted that the

ESTIMATING FRACTALDIMENSION WITH THE DIVIDER METHOD IN GEOMORPHOLOGY

measured length L (G) of coastlines varied depending upon the location of the starting point of the divider walk. If one considers the set of points at which a pair of dividers step on a line to be observations of the line's complexity, then any change in the starting point will give rise to a different set of observations. Variations in L (G) caused by changing the starting point of the divider walk could then be ascribed to sampling error. Changes in the starting point of a divider walk will not only affect L (G), but also estimates of D for the line. One solution to this problem is to increase the length of the line sampled. However, this is not usually possible inasmuch as most geomorphic phenomena are limited in extent. On a line of finite length, the number of steps N (i.e., sample size) is also finite for a given steplength G. Kent and Wong (1982) solved this problem by starting several divider walks from different points along a line and averaging the resulting values of L (G) to arrive at a more reliable estimate of D. Longley and Batty (1986), in a study examining the changes in fractal dimension of city boundaries through time, brought this solution to its logical conclusion by starting their divider walks from every point on the digitized line. This solution requires an enormous amount of computer time, and a smaller number of starting points should provide a sufficiently accurate estimate of fractal dimension• Longley and Batty ( 1986 ) were able to start their divider walks from any point on a line and still sample the entire line because the city boundaries they were studying are closed loops. On open-ended lines, the divider method must be modified to allow the use of multiple starting points while still sampling the entire length of the line. The divider program used in this analysis was written to accomplish this task. First, a starting point is selected at random. The dividers are then walked in either direction from that point to the opposite ends of the line. The total number of steps is then computed by simply adding the steps from the forward and

13 5

backward walks, including the last partial fractions. This procedure is then repeated for as many starting points as are desired. In order to determine the minimum number of starting points needed to obtain reliable estimates of D, an analysis was conducted using the four digitized lines shown in Fig. 1. Using the same set of steplengths throughout the analysis, values of D were estimated for each line using increasing numbers of starting points. For all four lines the standard deviation of each sample of D values was found to decline with increasing number of starting points. Results for one of the lines are shown in Fig. 3. In each case, the standard deviation of estimates of D for each starting point reached a minimum value before the number of starting points reached 50. Variations in estimates of D derived using 50 starting points occurred no higher than the fourth decimal place for all of the lines studied• Consequently, in the following analyses, 50 starting points were employed to estimate D. The effect of using mean values of L (G) to estimate D was also examined• Estimates of D derived from mean values of L (G) were compared with D values computed from regressions employing all of the original data (i.e., 50 values of L (G) for each G). Differences between the estimates of D generally occurred 0.014

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R. ANDRLE

only in the fourth decimal place, and at most did not exceed 0.003.

the associated t values ( t < - 3 7 1 . 9 7 ) for the four regressions (Table 1 ) would seem to indicate that the log N-log G relations are indeed linear and, hence, that the lines are statistically self-similar. However, lack of fit tests (Minitab Inc., 1985 ), which test for nonlinearity by conducting regressions on subsets of the data, reveal possible curvature in all of the relations. This curvature is clearly shown in plots of the standardized residuals from the four regressions (Fig. 5 ). Three of the four Richardson plots (shown in Fig. 6) appear linear, while the plot for the Grand Canyon contour line has a slight convex-upward curvature. Values of D calculated from the regressions of log L(G) against G (Table 1 ) duplicate those derived from the log N-log G plots. The r 2 and t values from the regressions were lower than those for the log N-log G plots but were also indicative of line-

Nonlinearity in Richardson and log N-log G plots In order to test the effect of nonlinearity in Richardson and log N-log G plots on estimating D for geomorphic lines, the divider program described above was applied to the four lines shown in Fig. 1. For each line, N was calculated for between 120 and 130 logarithmically distributed values of G. Both log N-log G and Richardson plots were constructed for each of the four lines. All four log N-log G plots (Fig. 4) appear linear. Therefore, least-squares linear regression analyses were performed (Table 1 ). D is equal to the absolute value of the slope of each of the relations. The high r 2 values (0.999) and ANTIETAM

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ESTIMATING FRACTALDIMENSIONWITH THE DIVIDER METHOD IN GEOMORPHOLOGY TABLE 1 Regressions of log N against log G for the four digitized lines Regression equation

r2

Degrees of freedom

t

Lack of fit test

0.999 0.986

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0.999 0.982

129 129

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0.999 0.975

119 119

-371.97 - 67.17

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0.999 0.983

129 129

- 1380.94 -86.67

P = 0.003 P = 0.003

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Grand Canyon contour lime log N = 7.18-1.22 log G log L(G) = 7.18-0.22 log G

Mindanao coastline log N = 7.20-1.07 log G log L(G) = 7.20-0.07 log G

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Fig. 6. Richardsonplots for the four digitized lines. arity. Lack of fit tests for the log L (G)-log G regressions indicated, as with the log N-log G plots, that the Richardson plots are nonlinear. Similarly, plots of the residuals from the regressions of log L(G) against log G (not shown for space considerations) were nearly identical to the patterns of residuals from the log N-log G plots shown in Fig. 5. Scatter in the residuals tends to increase with increasing G because N (analogous to sample size) decreases, resulting in increasing sampling error. This trend is an intrinsic characteristic of the divider method. Further investigation of the effect of nonlinearity in the Richardson and log N-log G plots on estimates of D was conducted. Ideally, for a line to be statistically self-similar over a given range of scale, D must be constant over the entire range of scale. If D is not constant, then estimates of D will be dependent upon the

number and range of G values used in their calculation. To test this, the 120 values of G and N from the log N-log G plot for Antietam Creek (Fig. 4) were partitioned successively into subsets of 10, 20, 30 and 40 values of G and N, and D was calculated for each subset. D was then plotted against scale of measurement S (equal to the geometric mean of G for each subset) (Fig. 7). Admittedly, the method of calculation of S in this manner is only an approximation, however, it is adequate for the purpose of this analysis. Progressive changes in the four plots in Fig. 7 demonstrate two sources of error in estimating D; sampling error, and model specification error. Increasing the number (and range ) of G values employed in the estimation of D reduces sampling error. This reduces error in estimating D (caused by sampling error ), as evidenced by the decrease in a m o u n t of random

ESTIMATING FRACTALDIMENSION WITH THE DIVIDER METHOD IN GEOMORPHOLOGY

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scatter in the D-log S plots with increasing number of steplengths. Clearly, in order to reduce sampling error it is desirable to use a greater number of steplengths in estimating D. However, as the range of steplengths used increases, error in estimates of D resulting from model specification error will increase if the Richardson or log N-log G plot is nonlinear. This is illustrated by the plot of D against log S for 40 steplengths (Fig. 7) where the convexupward curvature of the log N-log G relation (as shown by the residuals in Fig. 5) is reflected in the pattern of estimates of D. As the number of steplengths increases, overlap of the ranges of steplengths used to produce each estimate of D also increases. Thus, it is expected that with increasing range of measurement, variations in estimates of D resulting from differences in range of measurement should decline. This trend can also be seen in the D-log S plots (Fig. 7 ). As the ranges of G used in estimating each D value converge, estimates of

D will stabilize. From this analysis it appears that the best solution is to use a small range and large number of G values in order to minimize both sampling and model specification error in estimating D. Discussion

In searching for a reliable means of estimating D, one might argue that if a Richardson or log N-log G plot is found to be nonlinear, the range of scale of measurement is too great (i.e., the range of scale encompasses several fractal elements a n d / o r transition zones). Therefore, a solution which suggests itself would be to examine smaller ranges of scale until statistical tests for linearity show that the segment of the Richardson or log N-log G plot is linear. This linear segment (or fractal element) could then be considered evidence for statistical self-similarity over the range of scale encompassed by the fractal element. However, as shown by the

140 previous analysis, if the range o f scale is progressively reduced in this m a n n e r it is inevitable that increasing sampling error will eventually render insignificant any nonlinearity which exists in the log N-log G or log L ( G ) - l o g G relationship. This highlights the problem (which has not been adequately addressed in the literature) of defining the degree to which a line can deviate from statistical self-similarity before such deviations nullify usefulness of the concept.

Conclusions In this study, three sources of error in using the divider m e t h o d to estimate fractal dimension are investigated. Three possible solutions to the first source of error, accounting for the last partial step in a divider walk, are examined. First, the last partial step can be ignored. Second, the last partial step can be rounded off (i.e., another whole step is added if the last partial step is greater than G/2). Third, the distance to the endpoint of the digitized line can be added to the total n u m b e r of steps (expressed as a fraction o f G). Qualitative arguments support adding the last partial step as a fraction of G. However, more rigorous analysis of this problem is complicated by the dependence of this error on the configuration of the section o f line extending beyond the endpoint of the measured section. For this reason a definitive answer to this problem is not forthcoming. Nonetheless, it can be stated with certainty that the a m o u n t of error in estimates of D caused by this problem generally increases with increasing steplength. The second source of error, the starting point problem, has been addressed previously in the literature, and the solution employed in this study involves the use of multiple starting points and the calculation o f D from m e a n values of L (G). It was found that for the geomorphic lines used in this study, 50 starting points are sufficient to reduce error in estimates of D to a minimal level.

R.ANDRLE The third source of error (model specification error) results from the use of linear regression analysis in estimating the slope of nonlinear log N-log G or log L ( G )-log G relations. Only on strictly statistically self-similar lines will model specification error be zero. Typically, high r 2 values from regressions have been taken as sufficient evidence of statistical self-similarity in geomorphic research. However, given the results o f this study, researchers are advised to examine the residuals from regressions of log N or log L (G) against log G in order to determine the degree and nature of any nonlinearity which might exist. Several problems need to be addressed by future research. A standard m e t h o d of determining an acceptable level of error in estimating D by the divider m e t h o d does not currently exist, leaving the accuracy of such estimates open to question. Determining an acceptable level of error is further complicated by the lack of a precise definition of the requirements for statistical self-similarity (i.e., the degree to which a log N-log G or L ( G )-log G plot can be nonlinear and yet still be considered statistically self-similar).

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ESTIMATING FRACTAL DIMENSION WITH THE DIVIDER METHOD IN GEOMORPHOLOGY

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